J = 128πρν2 Z2
1
6 − 1
6(η2+ 1)3
!
. (2.32)
Figure 2.6: The momentum flux of a cylindrical jet, here plotted without the scaling 128πρν2/Z2.
It is seen from eq. (2.32) that the momentum flux falls as η to the sixth power. This is shown in figure 2.6 where the momentum flux converges very rapidly as η grows. As such, most of the momentum flux of the jet is con-tained within η <1. This might be seen as support of the assumption that the model of the jet will generally prove the best results for η <1.
We have not included any calculation of the volume flow for the laminar jet.
The reason for this is that the bulk of fluid that makes up the jet increases as more of the surrounding fluid is entrained onto the jet as it moves along the axis of motion. As such the momentum flux of the jet is the appropriate estimate for the size of the jet, as it remains the same for all values of Z.
2.4 Discussion
6
6We will in this subsection frequently refer to [1] D.J. Tritton ’Fluid Dynamics’
2.4.1 The edge of the jet
During operation the axial velocity is positive and non-zero inside the pipe system for all values ofRandZ. As such there will be a build up of a bound-ary layer along the entire pipe. Even so the axial velocity decreases rapidly for an increasing value of η, and as such it might be appropriate to define a value which we can expect a build up of a significant boundary layer on the inside of the pipe system containing the jet. We have previously argued that the model of the jet will likely prove the best results for η < 1. From eq.
(2.26) we have that the radial velocity reaches zero asη= 1. At this value of η it can be seen from eq. (2.28), that the axial velocity have reduced to one fourth of its maximum. As such, we might useη = 1⇔R =Z as a starting point for an experimental search for an appropriate value of η for which we can define to be the edge of the jet.
We also have that the assumption of a free laminar jet necessarily must be rendered invalid at some point because of the interference of the walls con-taining it. The error of this approximation will be omitted in this thesis, but having an edge of the jet enables us at least to measure the distance between the wall of the pipe system and what we regard as the jet. By choosingR =Z as the edge of the jet we see that we have defined that jet will propagate in a 45 degree angle in the radial direction as it moves along the axial direction.
2.4.2 The orifice from where the jet emerges
We will now discuss the consequences of utilizing the continuum equation in a model of a jet, and make clear the boundary conditions at the orifice from where the jet emerges.
As we substitute η = R/Z into eq. (2.28) and eq. (2.26), we have in the limit Z →0 respectively
w(R, Z) = lim
2.4. DISCUSSION 17 From eq. (2.33) and eq. (2.34) we see that there is no fluid entering into the system at Z = 0. The model therefore predicts that the jet is given momentum by a source emerging from an infinite small opening at (Z = 0, R = 0) that is not providing any mass to the system. This is of course unphysical, but it greatly simplifies the model. How the jet is given mass can be seen from dividing eq. (2.34) by eq. (2.33)
UR w =
−
4ν Z
(η2−1) (η2+1)2
ν Z
8 (η2+1)2
=−1
2(η2−1) = 1
2(1−R2
Z2). (2.35)
Eq. (2.35) shows that as we approachZ = 0 the radial velocity grows towards infinity for all values ofR. As such we find that the jet here is given its mass from fluid being drawn from the surroundings onto the the center just beyond Z = 0.
2.4.3 Stability
The velocity profile of a jet as a turning point i.e. a non-zero value for its second derivative. We know from experimental data that flows of this kind are much more prone to instabilities than flows without. Hence, we pre-sume that the jet will most likely dissolve close to the point from where it emerges. Nevertheless the surrounding walls of the pipe will serve to stabilize the jet, so that we might presume a jet structure of the flow for some length of the system. We will assume that the flow retains its characteristics as a jet at least long enough for its edge to reach the inner walls of the pipe i.e.
η= 1 ⇔R=Z.
2.4.4 The effects of fluctuations in the pressure gradi-ent on the jet
We presume that the effects that fluctuations of the pressure gradient in the flow will have upon both the jet and the surrounding fluid will be significant.
An increase of the velocity of the flow will lead to a higher degree of stability in the jet, while a decrease will make it more unstable. What effects rapid fluctuations will have on the flow near the valve will not be pursued further in this thesis. Nevertheless we will presume that the fluctuations will cause
a larger degree of instabilities in the jet, and that this might cause it to undergo the inevitable transition from a structured laminar jet to a regular turbulent flow at a faster rate than a stationary jet.
2.4.5 The transition from jet to regular turbulent flow
We presume that as the jet breaks down because of the interference with the wall or because of instabilities within the jet itself the flow will undergo a sudden transition from a laminar jet to a regular turbulent flow. We will presume it to be crucial that this transition has taken place i.e. that the flow has been stabilized enough for it to be described as a regular turbulent flow, before it reaches any point where measurements of it are obtained. It should be noted that it would possible prove very hard to obtain proper readings of the flow as it undergoes the transition, and that this might also prove right as we regard the jet as well.
Chapter 3
Turbulent Flow Near a Wall
3.1 Background
In this chapter we will presume the pipe to be filled with a more or less homogeneous turbulent flow. The high velocity of the flow required to make the assumptions for the jet valid, results in a high Reynolds number in the flow following the jet. As such we can not hope to find an analytic solution to the Navier-Stokes equation for the behavior of the flow. Instead we will rely on the analysis made by von Karman in his deduction of the behavior of a turbulent shear flow near a wall in order to describe the stream in this part of the system1.